Triangular number

The first six triangular numbers

A triangular number is a number that is the sum of all of the natural numbers up to a certain number. When formed using regularly spaced dots, they tend to form a shape of either an equilateral or a right triangle, hence the name.

For example, 10 is a "triangular number" because [math]\displaystyle{ 10=1+2+3+4 }[/math].

The first 25 triangular numbers are: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, and so on.

A triangular number is calculated by the equation: [math]\displaystyle{ \frac{n(n+1)}{2} }[/math].

Triangular Number Media

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  Plot of the tringular numbers.*Realised with Python and MatPlotLib.

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  Illustration of Triangular Number T 4 Leading to a Rectangle (yellow-green)

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  Square number 16 as sum of two triangular numbers

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  Square number 25 as sum of two triangular numbers

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  A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes. This shows that the square of the nth triangular number is equal to the sum of the first n cube numbers.

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  The fourth triangular number equals the third tetrahedral number as the nth k-simplex number equals the kth n-simplex number due to the symmetry of Pascal's triangle, and its diagonals being simplex numbers; similarly, the fifth triangular number (15) equals the third pentatope number, and so forth

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  Proof without words that the number of possible handshakes between n people is the (n−1)th triangular number